Page 1

Exact structured singular value of

robotic manipulators and

quantitative analysis of passivity based control

Satoru Sakai∗, Koichi Osuka†, Kenji Fujimoto‡,

∗Graduate School of Mechanical and Electrical Engineering

Chiba University, Inage-ku, Yayoi-cho 263–8522

Email: satorusakai@faculty.chiba-u.ac.jp

†Kobe University, Japan

‡Nagoya University, Japan

Telephone: (81) 43–290–3193, Fax: (81) 43–290–3196

Abstract—This paper gives an exact and explicit expression

of the structured singular value for robotic manipulators with

a passivity based control in port-controlled Hamiltonian form,

even though it is NOT possible to give the exact or explicit

structured singular value for general systems. First, we focus

on dynamics with endlink mass perturbation after the settling

time. Second, we derive the exact and explicit structured singular

value for manipulators by using structural properties of the

dynamics. The derived structured singular value is nothing but

the structured singular value of manipulators without control

because the passivity based control preserves the Hamiltonian

structure. Furthermore, based on the derived structured singular

value, we quantitatively analyze the robust stability of robotic

manipulators with the passivity based control.

I. INTRODUCTION

Hamiltonian control systems [16], [9] are the systems de-

scribed by Hamilton’s canonical equations which represent

general physical systems. Recently port-controlled Hamilto-

nian systems are introduced as a generalization of Hamiltonian

systems [6]. They can represent not only ordinary mechanical,

electrical and electro-mechanical systems (e.g.: magnetic lev-

itation system), but also nonholonomic systems (e.g.: vehicle

systems) [7], [5]. The special structure of physical systems

allows us to utilize the passivity which they innately possess

and a lot of fruitful results were obtained so far. These methods

are so called passivity based control [15], [11].

One of the advantages of passivity based control is ro-

bustness against physical parameter variations. Utilizing the

intrinsic passive property of physical systems, it is easy to

stabilize the system by using only the information of kinematic

parameters. It is usually difficult to stabilize a nonlinear

system without using dynamic parameters, such as, mass and

moment of inertia. Furthermore, not only the stabilization, but

also tracking and dynamic output feedback stabilization are

achieved in [1], [14] by the generalized canonical transfor-

mations [2]. These transformations are natural generalization

of the canonical transformations which are well-known in

classical mechanics and preserve the structure of the port-

controlled Hamiltonian systems.

In addition, some optimal gains are clarified to improve

transient behavior in the cases of a class of mechanical

Hamiltonian systems [14] and redundant manipulators [12],

[13]. These gains make the settling time smaller and are

still unclear for the other mechanical systems, electrical and

electro-mechanical systems.

On the other hand, when we control a robotic manipulators

in practice, these discussion can be useful only before the

settling time. After the manipulator endpoint converses to the

neighborhood of the desired point in free space, the endpoint

interacts with the environment which is not free space any

more. This means that the endlink mass is not constant and the

perturbation of the mass is unknown in the case of unstructured

environment. The discussion after the settling time is required

in the case of robotic manipulators apart from the other

Hamiltonian systems, such as magnetic levitation systems and

vehicle systems. There is few papers to quantitatively discuss

the robustness of the passivity based control after the settling

time.

In order to discuss this matter, the concept of the structured

singular value is needed. The structured singular value is

a well-known measure of not only the stability margin for

structured uncertainty but also robust performance.

However, unfortunately, the structured singular value can

not be expressed explicitly and not be calculated exactly in

general.

In this paper, we solve this problem and derive the explicit

and exact structured singular value of robotic manipulators.

In Section II, we focus on a dynamics with endlink mass

perturbation after the settling time. In Section III, as a main

result of this paper, we derive the explicit and exact structured

singular value by using the structural property of the dynamics,

In Section IV, quantitatively, we analyze the robust stability of

passivity based control. In Section V, we conclude this paper.

1-4244-0259-X/06/$20.00 ©2006 IEEE

2053

Proceedings of the 2006 IEEE/RSJ

International Conference on Intelligent Robots and Systems

October 9 - 15, 2006, Beijing, China

Page 2

II. PORT-CONTROLLED HAMILTONIAN SYSTEMS

A. Port-Controlled Hamiltonian Systems [17]

A port-controlled Hamiltonian system with a Hamiltonian

H is a system with the following state-space realization.

?

where u,y ∈ Rm, x ∈ Rnand J is skew-symmetric, i.e.

J = −JT. Port-controlled Hamiltonian systems are natural

generalization of physical systems. The following properties

of such systems are known.

Lemma 1 [17] Consider the system (1). Suppose Hamiltonian

H is lower bounded and satisfies ∂H/∂t ≤ 0. Then the system

is passive with respect to the storage function H, and the

following feedback renders (u,y) → 0. Furthermore, if the

system is zero-state detectable, then the following feedback

renders the system asymptotically stable

˙ x

y

=

=

J(x,t)∂H

g(x,t)T ∂H

∂x(x,t)T+ g(x,t)u

∂x(x,t)T

(1)

u = −Cy

(2)

where C > 0 is any positive definite matrix.

The zero-state detectability, which is assumed in Lemma

1, does not always hold for general systems. In such a case,

the stabilization method by generalized canonical transforma-

tion (which is a generalization of a stabilization method of

exploiting virtual potential energy [15]) is useful.

B. Generalized Canonical Transformation and Stabilization

[2]

A generalized canonical transformation is a set of trans-

formations in the following form which preserve the port-

controlled Hamiltonian structure as in (1).

Here, ¯ x, ¯H, ¯ y and ¯ u denote new state, new Hamiltonian,

new output and new input, respectively. This transformation

is a natural description of a canonical transformation which

is widely used for the analysis of Hamiltonian systems in

classical mechanics.

Theorem 1 [2] Consider the system (1). For any functions

U and β, there exists a pair of functions Φ and α such that

the set (3) yields a generalized canonical transformation and

the function α is given by α = gT ∂U

generalized canonical transformation if and only if

?

holds with a skew-symmetric matrix K?.

Furthermore transform the system (1) by the generalized

canonical transformation U and β such that H + U ≥ 0.

Then the new input-output mapping ¯ u ?→ ¯ y is passive with a

storage function¯H if and only if

+∂(H + U)

∂x

¯ x

¯H

¯ y

¯ u

=

=

=

=

Φ(x,t)

H(x,t) + U(x,t)

y + α(x,t)

u + β(x,t)

(3)

∂x

T. Further Φ yields a

∂Φ

∂(x,t)

J∂U

∂x

T+ gβ + K? ∂H+U

−1

∂x

T

?

= 0

(4)

∂H

∂xJ∂U∂x

T

g β −∂(H + U)

∂t

≥ 0.

(5)

q1

q2

Fig. 1.Two-link manipulator.

Suppose that (5) holds and that H + U is positive definite.

Then the feedback ¯ u = −K ¯ y with K > 0 renders the system

stable. Suppose moreover that the transformed system is zero-

state detectable with respect to x. Then the feedback renders

the system asymptotically stable.

C. Dynamics After Settling

The following Hamiltonian system is considered here.

with the Hamiltonian

?

˙ q

˙ p

?

=

?

0

I

−I 0

∂p(q,p)T

??

∂H

∂q

∂H

∂p

T

T

?

+

?0

I

?

u

y

=∂H

(6)

H

=

1

2pTM(q)−1p

(7)

where M(q) is mass matrix which makes this system equiv-

alent to equations of motion of robotic manipulators [8] and

q,p ∈ Rnare position and momentum, respectively.

By using the stabilization procedure based on Theorem 1,

the following controller renders (q,p) → (qd,0).

u = −Cy +∂U

∂q

T

.

(8)

where qdis the desired position and U is a positive definite

function with U = 0 at q = qd.

Here, we assume the moment of inertia is linear with respect

to the link mass (homogeneous link) and describe the case of

n = 2 shown in Fig.1 for simplicity. In addition, let U(q) =

−(K/2)(q − qd)2and K and C be diagonal without lose of

generality. From the following process in this paper, we can

see there are few difficulties to extend the results to the more

general cases.

After the settling time, the closed-loop system of (6), (7)

and (8), which is nothing but the classical Hamiltonian system,

?

˙ q

˙ p

?

=

?

0

I

−I C

?

∂(H+U)

∂q

∂(H+U)

∂p

T

T

≡ fcl(q,p)

(9)

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Page 3

with the Hamiltonian

H =

1

2pT

?

a + c + 2bcos(q2) c + bcos(q2)

c + bcos(q2)

c

?−1

p (10)

can be approximated as the following linearized system around

the desired point.

∂fcl

∂xx

≡

=

−ApK

where x = (q,p)Tand

?

−ac+b2cos2(qd2)

˙ x

=

Aclx

?

(11)

0

I2

−ApC

?

x

(12)

Ap=

c

−ac+b2cos2(qd2)

−(c+bcos(qd2))

−(c+bcos(qd2))

−ac+b2cos2(qd2)

a+c+2bcos(qd2)

−ac+b2cos2(qd2)

?

.

(13)

Remark Acl is NOT constant because the parameters in

Ap depend on the perturbation of the endlink mass m. For

example, in the case of two homogeneous rectangular links

with mass miand moment of inertia Ii, these parameters are

to the enter of mass. See [8] for the details of these physical

parameters.

a = I1+ I2+ m1r2

b = ml1r2

c = I2+ mr2

1+ m(l2

1+ r2

2)

2

(14)

where liis link length and ri is the distance form the joints

The above stabilization procedure is equivalent to the energy

shaping and damping injection [10]. However, we emphasize

the canonical transformation approach for two reasons. First,

we can clearly see that the passivity based control (8) preserves

Hamiltonian structure and the controlled manipulators are

nothing but new manipulators with spring and damping (9).

This gives an important interpretation to the following results

in this paper. Second, in our next work, we will extend the

following results to tracking and dynamic output feedback

stabilization cases which are the results due to the canonical

transformation approach.

Strictly speaking, the above linearization procedure should

be after the Legendre transformation: (q,p) → (q, ˙ q), because

the state p is the momentum and depends on the perturbation

of the endlink mass m. However results in this paper are the

same because Coriolis and centrifugal forces in ˙ p are high-

order nonlinear terms.

III. EXACT STRUCTURED SINGULAR VALUE

In this section, as the main result of this paper, we derive an

exact and explicit expression of the structured singular value of

the robotic manipulators with endlink mass perturbation after

the settling time.

In general, it is NOT possible to explicitly derive the

structured singular value. However, we overcome this problem

for robotics manipulators by using structural properties of the

dynamics.

A. Structured Singular Value

Consider the loop shown in Fig.2. For M

structured singular value µ is defined as

∈ Cn×n,

µ∆(M)=

1

min{¯ σ(∆)|∆ ∈ ∆, det(I − M∆)=0},

unless no ∆ ∈ ∆ makes I − M∆ singular, in which case

µ∆:= 0 , where

∆ = {diag(δ1Ir1,...,δSIrS,∆1,...,∆F)

s.t. δi∈ C,∆j∈ Cmj×mj},

(15)

(16)

S

?

i=1

ri+

F

?

j=1

mj= n.

(17)

Conceptually, the structured singular value is the stability

margin of M and nothing but a straightforward generalization

of the singular values for constant matrices. In other words,

the reciprocal of the largest singular value of M is a measure

of the smallest structured ∆ that causes instability of feedback

systems. In general, the function µ is not a norm, since it does

not satisfy the triangle inequality. However, for any a ∈ C,

µ(a) = ?a?µ(M), so in some sense, it is related to how “big”

the matrix is. The structured singular value is not only the

stability margin, which is related to transient behavior, but

also robust performance in robust control theory.

The following theorem and lemma are fundamental results

of robust linear control theory [18].

Theorem 2 Let b > 0. The loop shown in Fig.2 is well-posed

and internally stable for all ∆(•) ∈ M(∆) with ?∆?∞< 1/b

if and only if

supω∈Rµ∆(M(jω)) ≤ b

(18)

where ω is frequency and M(∆) is the set of all block

diagonal and stable rational transfer functions that have block

structures such as ∆.

Lemma 2 The structured singular value is related to the

following linear algebra quantities as

ρ(M) ≤ µ(M) ≤ ¯ σ(M))

(19)

where ρ(M) is the spectrum radius and ¯ σ(M) is the largest

singular value. The first equality holds at (S,F) = (1,0) and

the second equality holds at (S,F) = (0,1).

Unfortunately, in general, the structured singular value can

not be expressed explicitly though the lower bound and the

upper bounds in (19) can. Furthermore, the difference between

these bounds can be large arbitrary, so these bounds can not be

used for the estimation. In many practical cases, this difficulty

is avoided by numerical methods, even though, we can not yet

get the exact structured singular value because this numerical

computation is not a convex problem.

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B. Exact Structured Singular Value of Robotic Manipulators

In this subsection, we give an explicit and exact structured

singular value of the robotic manipulation system with endlink

mass perturbation after the settling time.

Theorem 3 Suppose the system in (12) has the following the

endlink mass perturbation,

m = ¯ m + δm

(20)

where ¯ m is the nominal endlink mass and δm is the corre-

sponding perturbation. Then, the structured singular value is

explicitly expressed as

????λi

where λ(•) is eigenvalue, ∆ = δmI, a = a1m+a2, b = b1m,

c = c1m,

?

?

µ∆(M) = maxi

??

A

C

B

D

??????

(21)

A =

0

I

−KAs

−CAs

?

(22)

B =

1

¯ mI

00

1

¯ mI

BsC

0

1

¯ mIBsK

BlI

0

00

0

BlI

0

0

0

BlI

?

(23)

C =

−A

I

−¯ mA

¯ mI

−¯ m2A

¯ m2I

−¯ m3A

¯ m3I

(24)

D=

−1

¯ mI

0

0

0

0

0

0

0

At

0

−BlI

0

−¯ mBlI

0

c1a1I

0

¯ mc1a1I

0

−Bt

0

−Bt

0

−¯ mBt

I

−¯ m2Bt

¯ mI

0

0

0

0

0

0

I

0

0

0

0

0

0

0

0

I

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

¯ mAt

I

¯ m2At

¯ mI

¯ m3At

¯ m2I

(25)

and

As=−1

me

?

−c1

b1cos(qd2) + c1

a2+ ¯ m(a1+c1+2bcos(qd2))

¯ mme

?

b1cos(qd2) + c1

?

(26)

At=

?

0

I

−KAl

−CAl

(27)

Al=−1

me

?

−c1

b1cos(qd2) + c1

a2+ ¯ m(a1+c1+2bcos(qd2))

c1a1−b2

b1cos(qd2) + c1

1cos2(qd2)

?

(28)

∆

M

Fig. 2.Closed-loop system with structured uncertainty.

Bs=

1

¯ mme

?

c1

−(b1cos(qd2) + c1)

a1+ c1+ 2bcos(qd2)

−(b1cos(qd2) + c1)

?

(29)

Bt=

?

0

0

Bl

0

?

(30)

Bl=a1c1− b2

1cos2(qd2)

¯ mme

(31)

me= a2c1+ ¯ m(a1c1− b2

1cos2(qd2)).

(32)

Before the proof of this theorem, we discuss the guideline.

In this proof, we focus on two special structures of the

closed-loop systems. First, the uncertainty of this robotic

manipulation system is structured, that is, the perturbation

exist in only the endlink mass, while the whole system is

physically modeled. This means that if we can pull out the

perturbation as ∆ = δmI, the first equality of (19) holds.

Now assume

Acl=¯Acl+ δm˜Acl.

(33)

In this case, intuitively, it is easy to give the corresponding

M(s) as

?

However the elements of Aclin (13) are not affine of m and

(33) does not hold. It is not trivial but difficult to derive the

corresponding M(s) to ∆. So, we need more detail structure

of the closed-loop system.

¯Acl+˜Acl

˜Acl

I

0

?

=˜Acl(sI − (¯Acl+˜Acl))−1.

(34)

Proof of Theorem 3.

We focus on a structure that the parameters in (13) are affine

of the endlink mass:

(a b c) = m(a1b1c1) + (a20 0),

(35)

because of homogeneous links. The elements of Acl are

written in rational polynomial form. This fact implies that

2056

Page 5

the corresponding descriptor form of (13) makes the elements

polynomial form with respect to m as follows,

Edes˙ x = Adesx

(36)

where

Edes

= {(−a1c1+ b2

≡

=

NAcl

?

1cos2(qd2))m2− a2c1m}I4(37)

NI4

Ades

(38)

=

0

NI

−KAdes(2,1)

−CAdes(2,1)

?

(39)

and

Ades(2,1)=

?

−c1m

(c1+ b1cos(qd2))m

(c1+b1cos(qd2))m (a1+c1+2b1cos(qd2))m+a2

?

.

In this case, the following relations hold [4],

[EdesAdes]=

Fu

?∆,

∆, S

¯

M?

=

Fu

???

0

P

I

Po

?

,

˜

M

??

where S(•,•) is the star product, Fu(•,•) is the lower linear

fractional transformation and

=

I,

Po

=¯ mI,

∆=

δmI.

P

(40)

Now, δms of Acl are successfully pulled out and the corre-

sponding M is given as follows.

M =

?

E−1

0A0

0A0+¯ M12b

E−1

0

¯

M21

−¯ M12aE−1

¯

M11−¯ M12aE−1

0

¯

M21

?

(41)

where E0,A0 are the corresponding nominal values of (36)

and

?

where the dimensions are omitted because of the uniqueness.

From here, it is a straightforward calculation to confirm

?

with (27), (23), (24) and (25).

Recall that ∆ is the repeated scalar block with only one

scalar and the first equality in the (19) holds. That is, the

structured singular value is expressed exactly and explicitly.

(Q.E.D.)

¯

M =

¯

¯

M11

M21

?¯

M12a¯

M22

M12b

?

¯

?

(42)

M =

A

C

B

D

?

(43)

Remark From the proof procedure, it is clear that this result

can be easily extended to the more general cases such as

n ≥ 3 with the other function U, such as that of gravity

compensator. These cases are omitted here only because of

the space limitation.

Theorem 3 is a very fundamental result in the field of

robotics. First, based on Theorem 3, we can study many

problems after the settling time, such as robust performance

in the presence of the perturbation. Second, the structured

singular value with the passivity based control is interpreted as

the the structured singular value of manipulator itself, because

the passivity based control (8) preserves the Hamiltonian

structure as we see in Section II.

IV. ROBUST STABILITY ANALYSIS

In this section, as a demonstrative application of Theorem 3,

we quantitatively analyze the robust stability of the passivity

based control. As we mentioned in Section I, the passivity

based control is qualitatively said to be robust and there

are some papers of experimental comparison between the

conventional control (e,g. computed torque control) and the

passivity based control [3]. However, there are few papers

to discuss the robust stability of the passivity based control

quantitatively and theoretically .

In general, unmodeled dynamics exists in high-frequency

region. However, in Theorem 3, the direct term D of transfer

function matrix M(s), is not zero, that is, M(s) is not strictly

proper. This means that the endlink mass perturbation can lose

the robust stability in the high-frequency region.

Now, we focus the following quantity.

M(∞) = D

(44)

based on the continuity of the structured singular value, µ :

Cn×n→ R.

Theorem 4 Consider the structured singular value (21) of

robotic manipulation system (13) after the settling time. Then,

the structured singular value is give as

µ∆(M(∞))

where

(45)

=

1/¯ m, cos2(qd2) ≤ r

???1/(¯ m + a2c1/(a1c1− b2

r =c1(2a1+ a2/¯ m)

1cos2(qd2))

???, cos2(qd2) > r

2b2

1

.

The size of square matrix D, that is, the order of the

characteristic equation of D is larger than four. Generally

we can not derive analytical eigenvalues of the corresponding

large matrices, because we can not solve the m-th (m > 4)

order algebra equations.

Numerical methods, such as QR factorization, do not solve

this problem in practical calculation time even in the two-link

case. (e.g. The readers can confirm this calculation time by

Maple 7.0).

So, again, we need to focus on structural properties of

robotic manipulators.

2057

Page 6

Proof of Theorem 4.

The structure of D is not triangular but has the same property:

det(D) = det([dij]) = Πn

i=1dii.

(46)

This is confirmed as the following. First, by using the follow-

ing equation, in repeat,

det(D) = det(Dii)det(Djj− DjiD−1

where the size of sub-matrix D∗∗is arbitrary and

?(1,2) (det(D22) ?= 0)

(47) is reduced to

?1

Then, from the structure of BlI, the property (46) is confirmed.

The eigenvalues of matrices with the property (46) is equal

to the diagonal components dii. Now, we can explicitly derive

the spectrum of D as

?

This means that spectrum radius is

jjDij)

(47)

(i,j) =

(2,1) (det(D11) ?= 0)

,

(48)

det(D) = det

¯ mI

?

det(−¯ mBlI)det(0).

(49)

σ(D)=0, −1

¯ m, −

1

¯ m + a2c1/(a1c1− b2

1cos2(qd2))

?

(50)

µ∆(M(∞))=ρ(D)

(51)

???

=max

?1

¯ m,

???

1

¯ m + a2c1/(a1c1− b2

1cos2(qd2))

?

The remaining parts, such as a calculation of r, are trivial and

the proof is straight forward. (Q.E.D.)

It is an interesting result that the structured singular value

µ∆(M(∞)) in (45) does not depend on the controller gains

K and C and determined by the only link parameter, even

though M(∞) in (44) depends on the gains.

This fact seems to point out “an importance of link design

(mechanical design) more than controller design in passivity

based control”, all the more because the passivity based control

utilizes physical properties of link dynamics more than the

other conventional controls (e.g. computed torque control).

From (45), not only the structured singular value µ∆(M(∞))

but also r can be new link design guidelines.

V. CONCLUSION

In this paper, we derive the explicit and exact structured

singular value of robotic manipulators. First, we focus on the

dynamics with endlink mass perturbation after the settling

time. Second, we derive the exact and explicit structured

singular value by using structural properties of the dynamics

even though the structured singular value can not be expressed

explicitly and exactly in general. Furthermore, quantitatively,

we analyze the robust stability of the passivity based control.

The authors believe that the derive structured singular value

is the very fundamental result in robotics.

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Natural resolution of ill-

VI. APPENDIX

Doyle’s notation:

?

A

C

B

D

?

= C(sI − A)−1B + D.

The star product:

?

where the upper linear fractional transformation: Fu(X,Y ) =

X22+X21Y (I−X11Y )−1X12and the lower linear fractional

transformation: Fl(X,Y ) = X11+ X12Y (I − X22Y )−1X21.

S(X,Y )=

Fl(X,Y11)

X12(I − Y11X22)−1Y12

Fu(Y,X22)

Y21(I − X22Y11)−1X21

?

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